We study here the behaviour of non-spacelike geodesics in dust collapse models in order to understand the casual structure of the spacetime. The geodesic families coming out, when the singularity is naked, corresponding to different initial data are worked out and analyzed. We also bring out the similarity of the limiting behaviour for different types of geodesics in the limit of approach to th...

– This note corrects and complements our paper entitled “Scattering matrices and scattering geodesics of locally symmetric spaces” (Ann. Sci. Éc. Norm. Sup. 34 (2001) 441–469).  2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Nous apportons des corrections et des compléments à notre article intitulé “Scattering matrices and scattering geodesics of locally symmetric spaces” paru...

In this paper we consider a dominating Finsler metric on a complete Riemannian manifold. First we prove that the energy integral of the Finsler metric satisfies the Palais-Smale condition, and ask for the number of geodesics with endpoints in two given submanifolds. Using Lusternik-Schnirelman theory of critical points we obtain some multiplicity results for the number of Finsler-geodesics betw...

We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H2-metric without zero order terms. We find isometries (called R-transforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give ...

This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Owen and Provan’s algorithm (2011) as a subroutine. Our algorithm is applicable to any CAT(0) space in which geodesics between two close points can be computed, not limit...

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for WeilPetersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson geodesics. As an application, we show the...

These notes accompany courses given in University of Bath and Northwestern University during their summer schools in 2016. The main topic is geodesics in first-passage percolation, specifically recent developments by Hoffman and by Damron-Hanson, on directional properties of infinite geodesics. As is done in our recent survey [6], we outline how many of the results from Damron-Hanson ’14 [10] c...

It is shown that the β-functions for four dimensional N = 2 supersymmetric YangMills theory without matter give integral curves on the moduli space some of which are geodesics of the natural metric on the moduli space. In particular the flow lines which cross-over from from the weak coupling limit (asymptotically free theory) to the singular points, representing the strong coupling limit, are g...

The main objective of this paper is to propose a new method to generate smooth interpolating curves on Stiefel manifolds. This method is obtained from a modification of the geometric Casteljau algorithm on manifolds and is based on successive quasi-geodesic interpolation. The quasi-geodesics introduced here for Stiefel manifolds have constant speed, constant covariant acceleration and constant ...

Projecting on a suitable subset of coordinates, a picture is constructed in which the conformal boundary of AdS 5 × S 5 and those of the plane wave resulting in the Penrose limit is located at the same line. All AdS 5 × S 5 and plane wave geodesics are classified in their integrated form. Performing the Penrose limit the approach of null geodesics reaching the conformal boundary of AdS 5 × S 5 ...